Electric Circuits

Current is the rate charge passes through a wire's cross section, $I = \frac{\Delta q}{\Delta t}$, measured in amperes. A potential difference, or emf, drives it; conventional current points the way positive charge would move, even though in metal wires it's electrons moving the other way.

Charge flows only around a closed loop; an open circuit has a break, and a short circuit is a path with almost no resistance, so charge crosses it without losing any potential. You read and draw circuits with standard schematic symbols for batteries, resistors, bulbs, switches, capacitors and meters, and one element can be part of more than one loop.

Resistance measures how hard it is for current to get through an element, and for a wire $R = \frac{\rho\ell}{A}$, where the resistivity ρ depends on the material (and for most conductors rises as they heat up). Ohmic elements follow $I = \frac{\Delta V}{R}$ with a constant resistance, so a graph of current against potential difference is a straight line with slope $\frac{1}{R}$.

Power is the rate an element transfers energy: $P = I\Delta V = I^2R = \frac{(\Delta V)^2}{R}$, in watts. A bulb's brightness rises with its power, so you can rank the bulbs in a circuit by comparing the power each one uses.

Resistors in series carry the same current and add, $R_{\text{eq}} = R_1 + R_2 + \cdots$. Resistors in parallel share the same potential difference and $\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$, so adding branches lowers the total resistance. A real battery's internal resistance lowers its terminal voltage when current flows, ammeters go in series and voltmeters go in parallel.

Kirchhoff's loop rule says the potential differences around any closed loop add up to zero, $\sum \Delta V = 0$, because a charge that goes all the way around ends with the energy it started with. A graph of potential against position around a loop shows it: the battery raises the potential and each resistor lowers it.

Kirchhoff's junction rule says the current into a junction equals the current out, $\sum I_{\text{in}} = \sum I_{\text{out}}$, because charge is conserved and doesn't pile up there. With the loop rule, it lets you find unknown currents in circuits with several branches.

Capacitors combine the opposite way to resistors: in parallel they add, $C_{\text{eq}} = C_1 + C_2 + \cdots$, and in series $\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots$, with each series capacitor holding the same charge. In an RC circuit an uncharged capacitor first acts like a wire, and after many time constants $\tau = RC$ its branch carries no current. In one time constant a charging capacitor reaches about 63% of its final charge (a discharging one drops to about 37%); you describe the change in between, but don't calculate it.